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Add implication-based rewritings from Babiak et al. (TACAS'12)

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* src/ltlvisit/simplify.cc: Implement them here, and augment them
to support M, and W operators.
* src/ltltest/reduccmp.test: Add some tests.
* doc/tl/tl.tex (Simplifications Based on Implications): Document
these rules.
* doc/tl/tl.bib (babiak.12.tacas): New entry.
This commit is contained in:
Alexandre Duret-Lutz 2012-01-10 19:34:41 +01:00
parent ed0dd0b48d
commit 212c7ebdd7
4 changed files with 124 additions and 5 deletions
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13
doc/tl/tl.bib
View file

@ -1,4 +1,17 @@
@InProceedings{ babiak.12.tacas,
author = {Thom{\'a}{\v{s}} Babiak and Mojm{\'i}r
K{\v{r}}et{\'i}nsk{\'y} and Vojt{\v{e}}ch {\v{R}e}eh{\'a}k
and Jan Strej{\v c}ek},
title = {{LTL} to {B\"u}chi Automata Translation: Fast and More
Deterministic},
year = 2012,
booktitle = {Proceedings of the 18th International Conference on Tools
and Algorithms for the Construction and Analysis of Systems
(TACAS'12)},
note = {To appear}
}
@InProceedings{ beer.01.cav,
author = {Ilan Beer and Shoham Ben-David and Cindy Eisner and Dana
Fisman and Anna Gringauze and Yoav Rodeh},

17
doc/tl/tl.tex
View file

@ -1472,8 +1472,8 @@ implication can be done in two ways:
In the following rewritings rules, $f\simp g$ means that $g$ was
proved to be implied by $f$ using either of the above two methods.
Additionally, implications denoted by $f\Simp g$ are only checked
if the ``\verb|ltl_simplifier_options::containment_checks_stronger|''
Additionally, implications denoted by $f\Simp g$ are only checked if
the ``\verb|ltl_simplifier_options::containment_checks_stronger|''
option is set (otherwise the rewriting rule is not applied).
\begin{equation*}
@ -1487,23 +1487,36 @@ option is set (otherwise the rewriting rule is not applied).
\text{if}& (\NOT f)\simp g &\text{then}& f\U g &\equiv \F g \\
\text{if}& f\simp g &\text{then}& f\U (g \U h) &\equiv g \U h \\
\text{if}& f\simp g &\text{then}& f\U (g \W h) &\equiv g \W h \\
\text{if}& g\simp f &\text{then}& f\U (g \U h) &\equiv f \U h \\
\text{if}& f\simp h &\text{then}& f\U (g \R (h \U k)) &\equiv g \R (h \U k) \\
\text{if}& f\simp h &\text{then}& f\U (g \R (h \W k)) &\equiv g \R (h \W k) \\
\text{if}& f\simp h &\text{then}& f\U (g \M (h \U k)) &\equiv g \M (h \U k) \\
\text{if}& f\simp h &\text{then}& f\U (g \M (h \W k)) &\equiv g \M (h \W k) \\
\text{if}& f\simp g &\text{then}& f\W g &\equiv g \\
\text{if}& (f\W g)\Simp g &\text{then}& f\W g &\equiv g \\
\text{if}& (\NOT f)\simp g &\text{then}& f\W g &\equiv \1 \\
\text{if}& f\simp g &\text{then}& f\W (g \W h) &\equiv g \W h \\
\text{if}& g\simp f &\text{then}& f\W (g \U h) &\equiv f \W h \\
\text{if}& g\simp f &\text{then}& f\W (g \W h) &\equiv f \W h \\
\text{if}& g\simp f &\text{then}& f\R g &\equiv g \\
\text{if}& g\simp \NOT f &\text{then}& f\R g &\equiv \G g \\
\text{if}& g\simp f &\text{then}& f\R (g \R h) &\equiv g \R h \\
\text{if}& g\simp f &\text{then}& f\R (g \M h) &\equiv g \M h \\
\text{if}& f\simp g &\text{then}& f\R (g \R h) &\equiv f \R h \\
\text{if}& h\simp f &\text{then}& (f\R g) \R h &\equiv g \R h \\
\text{if}& h\simp f &\text{then}& (f\M g) \R h &\equiv g \R h \\
\text{if}& g\simp f &\text{then}& f\M g &\equiv g \\
\text{if}& g\simp \NOT f &\text{then}& f\M g &\equiv \0 \\
\text{if}& g\simp f &\text{then}& f\M (g \M h) &\equiv g \M h \\
\text{if}& f\simp g &\text{then}& f\M (g \M h) &\equiv f \M h \\
\text{if}& f\simp g &\text{then}& f\M (g \R h) &\equiv f \M h \\
\text{if}& h\simp f &\text{then}& (f\M g) \M h &\equiv g \M h \\
\end{array}
\end{equation*}
The above rules were collected from various
sources~\cite{somenzi.00.cav,tauriainen.03.a83,babiak.12.tacas} and
sometimes generalized to support operators such as $\M$ and $\W$.
\appendix
\chapter{Syntactic Implications}\label{ann:syntimpl}

18
src/ltltest/reduccmp.test
View file

@ -1,5 +1,5 @@
#! /bin/sh
# Copyright (C) 2009, 2010, 2011 Laboratoire de Recherche et Developpement
#! /bin/sh
# Copyright (C) 2009, 2010, 2011, 2012 Laboratoire de Recherche et Developpement
# de l'Epita (LRDE).
# Copyright (C) 2004, 2006 Laboratoire d'Informatique de Paris 6 (LIP6),
# département Systèmes Répartis Coopératifs (SRC), Université Pierre
@ -174,6 +174,20 @@ for x in ../reduccmp ../reductaustr; do
run 0 $x '(a & b) M (a R c)' '(a & b)M c'
run 0 $x '(a & b) M (a M c)' '(a & b)M c'
run 0 $x 'a U ((a & b) U c)' 'a U c'
run 0 $x '(a&c) U (b R (c U d))' 'b R (c U d)'
run 0 $x '(a&c) U (b R (c W d))' 'b R (c W d)'
run 0 $x '(a&c) U (b M (c U d))' 'b M (c U d)'
run 0 $x '(a&c) U (b M (c W d))' 'b M (c W d)'
run 0 $x '(a R c) R (b & a)' 'c R (b & a)'
run 0 $x '(a M c) R (b & a)' 'c R (b & a)'
run 0 $x 'a W ((a&b) U c)' 'a W c'
run 0 $x 'a W ((a&b) W c)' 'a W c'
run 0 $x '(a M c) M (b&a)' 'c M (b&a)'
# Eventuality and universality class reductions
run 0 $x 'Fa M b' 'Fa & b'
run 0 $x 'GFa M b' 'GFa & b'

81
src/ltlvisit/simplify.cc
View file

@ -1272,9 +1272,16 @@ namespace spot
}
// if a => b, then a U (b U c) = (b U c)
// if a => b, then a U (b W c) = (b W c)
// if b => a, then a U (b U c) = (a U c)
// if a => c, then a U (b R (c U d)) = (b R (c U d))
// if a => c, then a U (b R (c W d)) = (b R (c W d))
// if a => c, then a U (b M (c U d)) = (b M (c U d))
// if a => c, then a U (b M (c W d)) = (b M (c W d))
if (b->kind() == formula::BinOp)
{
binop* bo = static_cast<binop*>(b);
// if a => b, then a U (b U c) = (b U c)
// if a => b, then a U (b W c) = (b W c)
if ((bo->op() == binop::U || bo->op() == binop::W)
&& c_->implication(a, bo->first()))
{
@ -1282,6 +1289,32 @@ namespace spot
result_ = b;
return;
}
// if b => a, then a U (b U c) = (a U c)
if (bo->op() == binop::U
&& c_->implication(bo->first(), a))
{
result_ = recurse_destroy
(binop::instance(binop::U,
a, bo->second()->clone()));
b->destroy();
return;
}
// if a => c, then a U (b R (c U d)) = (b R (c U d))
// if a => c, then a U (b R (c W d)) = (b R (c W d))
// if a => c, then a U (b M (c U d)) = (b M (c U d))
// if a => c, then a U (b M (c W d)) = (b M (c W d))
if ((bo->op() == binop::R || bo->op() == binop::M)
&& bo->second()->kind() == formula::BinOp)
{
binop* cd = static_cast<binop*>(bo->second());
if ((cd->op() == binop::U || cd->op() == binop::W)
&& c_->implication(a, cd->first()))
{
a->destroy();
result_ = b;
return;
}
}
}
break;
@ -1317,13 +1350,30 @@ namespace spot
if (bo->op() == binop::R
&& c_->implication(a, bo->first()))
{
b->destroy();
result_ = recurse_destroy
(binop::instance(binop::R, a,
bo->second()->clone()));
b->destroy();
return;
}
}
if (a->kind() == formula::BinOp)
{
// if b => a then (a R c) R b = c R b
// if b => a then (a M c) R b = c R b
binop* bo = static_cast<binop*>(a);
if ((bo->op() == binop::R || bo->op() == binop::M)
&& c_->implication(b, bo->first()))
{
result_ = recurse_destroy
(binop::instance(binop::R,
bo->second()->clone(),
b));
a->destroy();
return;
}
}
break;
case binop::W:
@ -1347,9 +1397,12 @@ namespace spot
}
// if a => b, then a W (b W c) = (b W c)
// (Beware: even if a => b we do not have a W (b U c) = b U c)
// if b => a, then a W (b U c) = (a W c)
// if b => a, then a W (b W c) = (a W c)
if (b->kind() == formula::BinOp)
{
binop* bo = static_cast<binop*>(b);
// if a => b, then a W (b W c) = (b W c)
if (bo->op() == binop::W
&& c_->implication(a, bo->first()))
{
@ -1357,6 +1410,17 @@ namespace spot
result_ = b;
return;
}
// if b => a, then a W (b U c) = (a W c)
// if b => a, then a W (b W c) = (a W c)
if ((bo->op() == binop::U || bo->op() == binop::W)
&& c_->implication(bo->first(), a))
{
result_ = recurse_destroy
(binop::instance(binop::W,
a, bo->second()->clone()));
b->destroy();
return;
}
}
break;
@ -1400,6 +1464,21 @@ namespace spot
return;
}
}
if (a->kind() == formula::BinOp)
{
// if b => a then (a M c) M b = c M b
binop* bo = static_cast<binop*>(a);
if (bo->op() == binop::M
&& c_->implication(b, bo->first()))
{
result_ = recurse_destroy
(binop::instance(binop::M,
bo->second()->clone(),
b));
a->destroy();
return;
}
}
break;
}
trace << "bo: no inclusion-based rules matched" << std::endl;